The group of rational points on the unit circle.9 A point in thecomplex plane with both coordinates rational is called a rational point.The set of rational points on the unit circle will be denoted by C(Q):C(Q) = {a + bi C | a, b € Q, and a² + b2 = 1}.(a) Show that, with the usual multiplication of complex numbers, C(Q)is a group.(b) Is i e C(Q)? What is o(i)?(c) Let x =+i. Is x E C(Q)? Find x², x³, and x-1.(d) A triple (a, b, c), where a, b, and c are integers, c + 0, and a²+b² = c²is called a Pythagorean triple. Given an element of C(Q), can youconstruct a Pythagorean triple? Given a Pythagorean triple, can youconstruct an element of C(Q)? Under this correspondence, when dotwo different Pythagorean triples give the same element of C(Q)?(e) In C(Q), what is the order of 3/5 + 4/5i? Make a conjecture.3

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Answered: The group of rational points on the…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.3: De Moivre’s Theorem And Roots Of Complex Numbers

Problem 16E

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